To rewrite the expression [tex]\(\left(\log \frac{1}{y^8}\right)^2\)[/tex] in terms of [tex]\(v\)[/tex] given that [tex]\(v = \log y\)[/tex], we can follow these steps:
1. Use the properties of logarithms:
[tex]\[
\log \left(\frac{1}{y^8}\right) = \log 1 - \log(y^8)
\][/tex]
We know that [tex]\(\log 1 = 0\)[/tex]. So the expression simplifies to:
[tex]\[
\log 1 - \log(y^8) = -\log(y^8)
\][/tex]
2. Apply the power rule of logarithms:
[tex]\[
-\log(y^8) = -8 \log(y)
\][/tex]
3. Substitute [tex]\(v\)[/tex] for [tex]\(\log(y)\)[/tex]:
[tex]\[
-8 \log(y) = -8 v
\][/tex]
4. Square the expression:
[tex]\[
\left(-8 v\right)^2
\][/tex]
5. Simplify the squared expression:
[tex]\[
(-8 v)^2 = 64 v^2
\][/tex]
Thus, the expression [tex]\(\left(\log \frac{1}{y^8}\right)^2\)[/tex] in terms of [tex]\(v\)[/tex] is:
[tex]\[
64 v^2
\][/tex]
